This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Plane Elasticity – Governing Equations”.

1. In solid mechanics, what does linearized elasticity deal with?

a) Small deformations in linear elastic solids

b) Large deformations in linear elastic solids

c) Large deformations in non-Hookean solids

d) Small deformations in non-Hookean solids

View Answer

Explanation: The part of solid mechanics that deals with stress and deformation of solid continua is called Elasticity. Linearized elasticity is concerned with small deformations (i.e., strains and displacements that are very small compared to unity) in linear elastic solids or Hookean solids (i.e., obey Hooke’s law).

2. For plane elasticity problems in three dimensions, which option is not responsible for making the solutions independent of one of the dimensions?

a) Geometry

b) Boundary conditions

c) Externally applied loads

d) Material

View Answer

Explanation: Elasticity is the part of solid mechanics that deals with stress and deformation of solid continua. There is a class of problems in elasticity whose solution (i.e., displacements and stresses) is not dependent on one of the coordinates because of their geometry, boundary conditions, and externally applied loads. Such problems are called plane elasticity problems.

3. For a plane strain problem, which strain value is correct if the problem is characterized by the displacement field u_{x}=u_{x}(x,y), u_{y}=u_{y}(x,y) and u_{z}=0?

a) ε_{xy}=0

b) ε_{xz}=0

c) ε_{yz}≠0

d) ε_{xz}≠0

View Answer

Explanation: The plane strain problems are characterized by the displacement field u

_{x}=u

_{x}(x,y), u

_{y}=u

_{y}(x,y) and u

_{z}=0, where (u

_{x}, u

_{y}, u

_{z}) denote the components of this displacement vector u in the (x, y, z) coordinate system. The displacement field results in the following strain field:

\(\epsilon_{xz}=\epsilon_{yz}=\epsilon_{zz}=0, \epsilon_{xx}=\frac{\partial u_x}{\partial x}, 2\epsilon_{xy}=\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\) and \(\epsilon_{yy}=\frac{\partial u_y}{\partial y}\).

4. Underplane strain condition, what is the value of ε_{yy} if the problem is characterized by the displacement field u_{x}=2x+3y, u_{y}=5y^{2}, and u_{z}=0?

a) 10y

b) 5y

c) 3

d) 0

View Answer

Explanation: The plane strain problems are characterized by the displacement field u

_{x}=u

_{x}(x,y), u

_{y}=u

_{y}(x,y) and u

_{z}=0, where (u

_{x}, u

_{y}, u

_{z}) denote the components of the displacement vector u in the (x, y, z) coordinatesystem. The displacement field results in the following strain field:

\(\epsilon_{xz}=\epsilon_{yz}=\epsilon_{zz}=0, \epsilon_{xx}=\frac{\partial u_x}{\partial x}, 2\epsilon_{xy}=\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\) and \(\epsilon_{yy}=\frac{\partial u_y}{\partial y}\)

Thus, \(\epsilon_{yy}=\frac{\partial 5y^2}{\partial y}\)

=5*\(\frac{\partial y^2}{\partial y}\)

=5*2y

=10y.

5.For a plane strain problem, the relation between stress and strain components for an orthotropic material is σ=Cε. Which option is the correct structure of the matrix C?

a) \(\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}\)

b) \(\begin{pmatrix}\bar{c}_{11} & 0 & \bar{c}_{13} \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}\)

c) \(\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\-\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}\)

d) \(\begin{pmatrix}\bar{c}_{11} & -\bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}\)

View Answer

Explanation: For an orthotropic material under plane strain, with principal material axes (x

_{1}, x

_{2}, x

_{3}) coinciding with the (x, y, z) coordinates, the relation between stress and strain components is \(\begin{pmatrix}\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}\end{pmatrix} = \begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix} = \begin{pmatrix}\varepsilon_{xx} \\ \varepsilon_{yy} \\ 2 \varepsilon_{xy}\end{pmatrix}\) where C is the elastic stiffness matrix. The state of stress is σ

_{xz}=σ

_{yz}=0 and \(\sigma_{zz}=E_3(\frac{v_{13}}{E_1}\sigma_{xx}+\frac{v_{23}}{E_2}\sigma_{yy})\).

6. For an orthotropic material, if E and v represent Young’s modulus and the poisons ratio, respectively, then what is the value of v_{12} if E_{1}=200 Gpa, E_{2}=160 Gpa and v_{21}=0.25?

a) 0.3125

b) 0.05

c) 0.2125

d) 0.3

View Answer

Explanation: For an orthotropic material, E

_{1}and E

_{2}are the principal (Young’s) moduli in the x and y directions, respectively. The poisons ratio and Young’s moduli are related by the equation

v

_{12}=v

_{21}\(\frac{E_1}{E_2}\).

v

_{12}=0.25*\(\frac{200}{160}\)

=0.25*1.25

=0.3125.

7. Under plane stress condition in the XYZ Cartesian system, which stress value is correct if a problem is characterized by the stress field σ_{xx}=σ_{xx}(x,y), σ_{yy}=σ_{yy}(x,y) and σ_{zz}=0?

a) σ_{xy}=0

b) σ_{yx}≠0

c) σ_{zx}≠0

d) σ_{yz}≠0

View Answer

Explanation: A state of plane stress in XYZ Cartesian system is defined as one in which the following stress field exists:

σ

_{xz}=σ

_{yz}=σ

_{zz}=0, σ

_{xx}(x,y), σ

_{xy}=σ

_{xy}(x,y) and σ

_{yy}=σ

_{yy}(x,y).

Thus, σ

_{xx}, σ

_{xy}and σ

_{yy}are non-zero stresses. Such a problem in three dimensions can be dealt with as a two-dimensional (plane) problem.

8. For theplane stress problem in XYZ Cartesian system, σ_{xx}=σ_{xx}(x,y), σ_{yy}=σ_{yy}(x,y) and σ_{zz}=0, which option is correct regarding the associated strain field?

a) ε_{xx}=0

b) ε_{yx}=0

c) ε_{zx}=0

d) ε_{yy}=0

View Answer

Explanation: The strain field associated with the given stress field has the form ε=Sσ, where the matrix S is a symmetric matrix, and it is called elastic compliances matrix. In the XYZ Cartesian system, all the strain components except ε

_{yz}and ε

_{zx}are non-zero. Thus, ε

_{xx}≠0, ε

_{yy}≠0, ε

_{zz}≠0, ε

_{xy}≠0, where as ε

_{yz}=0 and ε

_{zx}=0.

9. For any two cases of plane elasticity problems, if the constitutive equations are different, then their final equations of motion are also different.

a) True

b) False

View Answer

Explanation: The equations of motion for plane elasticity problems are given by D*σ+f=ρü in the vector form, where f denotes body force vector, σ is the stress vector, u is displacement vector, D is a matrix of the differential operator, and ρ is the density. Note that the equations of motion of plane stress and plane strain cases differ from each other only on account of the difference in their constitutive equations.

10. In solid mechanics, which option is not a characteristic of a plane stress problem in the XYZ Cartesian system?

a) One dimension is very small compared to the other two dimensions

b) All external loads are coplanar

c) Strain along any one direction is zero

d) Stress along any one direction is zero

View Answer

Explanation: An example of a plane stress problem is provided by a plate in the XYZ Cartesian system that is thin along the Z-axis. It is acted upon by external loads lying in the xy plane (or parallel to it) that are independent of the Z coordinate. Thus, stresses and strains are observed in all directions except that the stress is zero along the Z-axis.

11. In solid mechanics, what is the correct vector form of the equations of motion for a plane elasticity problem?

a) D*σ+f=ρü

b) D*σ+f=ρu̇

c) D^{2}*σ+f=ρü

d) D*σ+f=ρu

View Answer

Explanation: For plane elasticity problems, the equations of motion are one of the governing equations. The vector form of equations of motion is D*σ+f=ρü, where f denotes body force vector, σ is the stress vector, u is the displacement vector, D is a matrix of differential operator and ρ is the density.

12. For plane elasticity problems, which type of boundary condition is represented by the equation t_{x}≡σ_{xx}n_{x}+σ_{xy}n_{y}, where t_{x} is surface traction force and n is direction cosine?

a) Essential boundary condition

b) Natural boundary condition

c) Both Essential and natural boundary conditions

d) Dirichlet boundary condition

View Answer

Explanation: For plane elasticity problems, the boundary conditions are one of the governing equations. There are two types of boundary conditions, namely, essential boundary conditions and natural boundary conditions. The equation t

_{x}≡σ

_{xx}n

_{x}+σ

_{xy}n

_{y}represents natural boundary condition or Neumann boundary condition.

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